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MATHEMATICS - CASE STUDY SOLUTION

y0
1.5
y5
4.4
y10
3.8
y1
3.5
y6
3.5
y11
3.5
y2
4.8
y7
2.4
y12
1.5
y3
5.3
y8
3.1
y4
5.0
y9
3.6

Kidney-shaped Area
 

A kidney-shaped area needs to be determined with the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule. The width is measured at 1 foot interval.

Since the area under a curve equals the definite integral of the curve function, the area can be approximated by using the rule of approximate integration.

 

  1. Midpoint Rule

Midpoint Rule Approximation
 

The Midpoint Rule approximation is

      Area = Δx(y0 + y1 + y2 + ... + y12)
             = 1(1.5 + 3.5 +4.8 + ... + 1.5)
             = 45.9 ft2

  2. Trapezoidal Rule

Trapezoidal Rule Approximation
 

The Trapezoidal Rule approximation is

      Area = Δx/2(y0 + 2y1 +2 y2 + ... + 2y11+ y12)
             = 1/2(1.5 +2(3.5) + 2(4.8) + ... + 2(3.5) + 1.5)
             = 44.4 ft2

 

3. Simpson's Rule


Simpson's Rule Approximation
 

The Simpson's Rule approximation is

      Area = h/3(y0 + 4y1 +2 y2 + ... + 4y11+ y12)
            = 1/3(1.5 + 4(3.5) + 2(4.3) + ... +4(3.5) + 1.5)
            = 44.73 ft2